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    <title>标架与曲面论基本定理</title>
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<h2>活动标架</h2>

<h3>向量场, 活动标架</h3>

<p class="definition">
	设 `S: bm r(u,v)` 是 `E^3` 的曲面. 称 `bm x(u,v)` 是 `S`
	上的光滑<b>向量场</b>, 如果对任意一点 `bm r(u, v) in S`, `bm x(u, v)`
	是从该点出发的一个向量, 并且 `bm x(u,v)` 光滑地依赖于参数 `(u,v)`.
	对任意 `(u, v)`, 当 `bm x(u, v)` 是 `S` 在 `bm r(u, v)` 的切向量时,
	称为<b>切向量场</b>; 当 `bm x(u, v)` 是 `S` 在 `bm r(u,v)`
	的法向量时, 称为<b>法向量场</b>. 例如, `bm r_u, bm r_v` 是切向量场,
	`bm n` 是法向量场 (视它们为 `(u,v)` 的函数).
</p>

<p class="definition">
	称坐标系
	<span class="formula">
		`{bm r(u,v)";" bm x_1(u,v), bm x_2(u,v), bm x_3(u,v)}`,
		`(bm x_1, bm x_2, bm x_3) != bb 0`, `AA (u,v) in D`
	</span>
	为曲面 `S` 上的<b>活动标架 (Movable Frame)</b>.
	一般要求 `(x_1, x_2, x_3) gt 0`
	以保证这组标架为正定向的. 特别当 `{bm x_1, bm x_2, bm x_3}`
	为单位正交标架时, 称为曲面 `S` 的<b>正交活动标架</b>,
	简称<b>正交标架</b>. `{bm r(u,v)";" bm r_u, bm r_v, bm n}`
	是活动标架的一个例子, 称为<b>自然标架</b>.
</p>

<h2>自然标架的运动方程</h2>

<h3>张量记号</h3>

<span class="formula">
	`u^1 = u`, `u^2 = v`;<br/>
	`bm r_alpha = (del bm r)/(del u^alpha)`, `alpha = 1, 2`;<br/>
	`bm r_(alpha beta) = (del^2 bm r)/(del u^alpha del u^beta)`,
	`alpha, beta = 1, 2`;<br/>
	`bm n = (bm r_1 ^^ bm r_2)/|bm r_1 ^^ bm r_2|`;<br/>
	`g_(alpha beta) = (:bm r_alpha, bm r_beta:)`, `alpha,beta = 1,2`;<br/>
	`b_(alpha beta) = (:bm r_(alpha beta), bm n:) = -(:bm r_alpha, bm
	n_beta:)`, `alpha,beta = 1,2`;<br/>
</span>

<p>	在这些记号下, `(g_(alpha beta)) = [E,F; F,G]`, `(b_(alpha beta)) =
	[L,M; M,N]`, 且 `g_(alpha beta) = g_(beta alpha)`, `b_(alpha beta)
	= b_(beta alpha)`. 我们又记
	<span class="formula">
		`(g^(alpha beta)) = (g_(alpha beta))^-1`,<br/>
		`b_alpha^beta = b_(alpha gamma) g^(gamma beta)`,<br/>
		`(delta_alpha^beta) = bm E`.
	</span>
	即 `g^(alpha beta)` 是 `(g_(alpha beta))` 的逆矩阵相应位置的元素,
	`(b_alpha^beta)` 是 Weingarten 变换在 `bm r_u, bm r_v` 下的矩阵,
	`delta_alpha^beta` 是 Kroneker 符号.
	我们有 `g^(alpha beta) = g^(beta alpha)`, 但一般 `b_alpha^beta !=
	b_beta^alpha`.
</p>

<h3>Einstein 求和约定</h3>

<p>	在一个单项式中, 若一个指标字母 (如 `alpha`) 作为上标和下标各出现一次,
	则该式表示对 `alpha = 1, 2` 的求和式;
	上下指标多对重复出现就表示该式是多重求和式. 如
	<span class="formula">
		`"d"bm r = bm r_1 "d"u^1 + bm r_2 "d"u^2 = bm r_alpha "d"u^alpha`,
		<br/>
		`"I" = g_(11)"d"u^1"d"u^1 + 2g_(12)"d"u^1"d"u^2 +
		g_(22)"d"u^2"d"u^2 = g_(alpha beta)"d"u^alpha"d"u^beta`,<br/>
		`"II" = b_(11)"d"u^1"d"u^1 + 2b_(12)"d"u^1"d"u^2 +
		b_(22)"d"u^2"d"u^2 = b_(alpha beta)"d"u^alpha"d"u^beta`.
	</span>
</p>

<h3>自然标架的运动方程</h3>

<p class="theorem">
  用待定系数法推导曲面 `S` 自然标架 `{bm r";" bm r_1, bm r_2, bm n}`
	的运动方程:
	<span class="formula">`{
		(del bm r)/(del u^alpha) = bm r_alpha, alpha = 1","2;
		(del bm r_alpha)/(del u^beta)
			= Gamma_(alpha beta)^gamma bm r_gamma + b_(alpha beta) bm n,
			alpha"," beta = 1"," 2;
		(del bm n)/(del u^alpha) = -b_alpha^beta bm r_beta,
			alpha = 1","2
	:}`</span>
  其中
  <span class="formula">
    `Gamma_(alpha beta)^gamma`
		`= 1/2 g^(xi gamma)(
			(del g_(alpha xi))/(del u^beta)
		+	(del g_(beta xi))/(del u^alpha)
		-	(del g_(alpha beta))/(del u^xi) )`.
		<span class="label" id="christoffel1"></span>
  </span>
	可以看到, 自然标架的运动由第一, 第二基本形式的系数完全决定.
</p>

<p class="proof">
	设
	<span class="formula">
		`(del bm r_alpha)/(del u^beta) = bm r_(alpha beta) = Gamma_(alpha
		beta)^gamma bm r_(gamma) + C_(alpha beta) bm n`,
		<span class="label" id="natural-eq1"></span>
	</span>
	<span class="formula">
		`(del bm n)/(del u^alpha) = bm n_alpha = D_alpha^beta bm r_beta +
		D_alpha bm n`.
		<span class="label" id="natural-eq2"></span>
	</span>
	<a class="ref" href="#natural-eq1"></a> 两边与 `bm n` 作内积得
	<span class="formula">
		`C_(alpha beta) = (:bm r_(alpha beta), bm n:) = b_(alpha beta)`.
	</span>
	<a class="ref" href="#natural-eq2"></a> 两边与 `bm n` 作内积得
	<span class="formula">
		`D_alpha = (:bm n_alpha, bm n:) = 0`.
	</span>
	<a class="ref" href="#natural-eq2"></a> 两边与 `bm r_gamma` 作内积得
	<span class="formula">
		`D_alpha^beta (:bm r_beta, bm r_gamma:) = (:bm n_alpha, bm
		r_gamma:) = -b_(alpha gamma)`.
	</span>
	上式两边同乘 `g^(gamma xi)`, 并对 `gamma` 求和, 有
	<span class="formula">
		` D_alpha^xi
		= D_alpha^beta delta_beta^xi
		= D_alpha^beta g_(beta gamma) g^(gamma xi)
		= -b_(alpha gamma) g^(gamma xi)
		= -b_alpha^xi`.
	</span>
	最后, 分别对 `g_(alpha beta)`, `g_(alpha gamma)`, `g_(beta gamma)`
	求偏导有
	<span class="formula">
		`(del g_(alpha beta))/(del u^gamma)
		= (:bm r_(alpha gamma), bm r_beta:)
		+ (:bm r_(beta gamma), bm r_alpha:)`,<br/>
		`(del g_(alpha gamma))/(del u^beta)
		= (:bm r_(alpha beta), bm r_gamma:)
		+ (:bm r_(gamma beta), bm r_alpha:)`,<br/>
		`(del g_(beta gamma))/(del u^alpha)
		= (:bm r_(beta alpha), bm r_gamma:)
		+ (:bm r_(gamma alpha), bm r_beta:)`.
	</span>
	后两式相加, 减去第一式得
	<span class="formula">
		`(:bm r_(alpha beta), bm r_gamma:) = 1/2 (
			(del g_(alpha gamma))/(del u^beta)
		+	(del g_(beta gamma))/(del u^alpha)
		-	(del g_(alpha beta))/(del u^gamma) )`.
	</span>
	现在 <a class="ref" href="#natural-eq1"></a> 两边与 `bm r_xi` 作内积得
	<span class="formula">
		`Gamma_(alpha beta)^gamma g_(gamma xi) = (:bm r_(alpha beta), bm
		r_xi:)`,
	</span>
	于是
	<span class="formula">
		` Gamma_(alpha beta)^gamma
		= Gamma_(alpha beta)^eta delta_eta^gamma
		= Gamma_(alpha beta)^eta g_(eta xi) g^(xi gamma)
		= (:bm r_(alpha beta), bm r_xi:) g^(xi gamma)`
		`= 1/2 g^(xi gamma)(
			(del g_(alpha xi))/(del u^beta)
		+	(del g_(beta xi))/(del u^alpha)
		-	(del g_(alpha beta))/(del u^xi) )`.
	</span>
</p>

<p class="definition">
	`Gamma_(alpha beta)^gamma` 称为 <b>(第一类) Christoffel 符号</b>,
	它由曲面第一基本形式的系数及其偏导数完全决定. 又
	<span class="formula">
		` Gamma_(xi alpha beta)`
    `= g_(gamma xi) Gamma_(alpha beta)^gamma`
		`= (:bm r_(alpha beta), bm r_xi:)`
		`= 1/2 (
			(del g_(alpha xi))/(del u^beta)
		+	(del g_(beta xi))/(del u^alpha)
		-	(del g_(alpha beta))/(del u^xi) )`
	</span>
	称为曲面的<b>第二类 Christoffel 符号</b>.
  可以看出, 两类 Christoffel 符号都关于 `alpha, beta` 对称.
</p>

<p class="corollary">
    <span class="formula">
      ` Gamma_(delta eta gamma) - (del g_(delta eta))/(del u^gamma)
      = 1/2 (
        - (del g_(eta delta))/(del u^gamma)
        + (del g_(gamma delta))/(del u^eta)
        - (del g_(eta gamma))/(del u^delta)
        )
      = -Gamma_(eta delta gamma)`,
      <span class="label" id="christoffel2"></span>
    </span>
    <span class="formula">
      `b_beta^xi Gamma_(xi alpha gamma)`
      `= b_(beta delta) g^(delta xi) g_(xi eta) Gamma_(alpha gamma)^eta`
      `= b_(beta eta) Gamma_(alpha gamma)^eta`.
    </span>
</p>

<h3>Christoffel 符号的表达式</h3>

<p>	由 <a class="ref" href="#christoffel1"></a> 式:
	<span class="formula">
		` Gamma_(alpha beta)^gamma
		= 1/2 g^(xi gamma)(
			(del g_(alpha xi))/(del u^beta)
		+	(del g_(beta xi))/(del u^alpha)
		-	(del g_(alpha beta))/(del u^xi) )`
	</span>
	可以求出
	<span class="formula">
		`[Gamma_11^1, Gamma_12^1; Gamma_21^1, Gamma_22^1]`
		`= 1/(2(EG-F^2)) [G E_u + F E_v - 2F F_u, G E_v - F E_v;
		   G E_v - F E_v, 2G F_v - G G_u - F G_v]`
		`overset ** =
		  1/2 [(ln E)_u, (ln E)_v; (ln E)_v, -G_u//E]`,<br/>

		`[Gamma_11^2, Gamma_12^2; Gamma_21^2, Gamma_22^2]`
		`= 1/(2(EG-F^2)) [2E F_u - F E_u - E E_v, E G_u - F E_v;
		   E G_u - F E_v, F G_u + E G_v - 2F F_v]`
		`overset ** =
		  1/2 [-E_v//G, (ln G)_u; (ln G)_u, (ln G)_v]`.
	</span>
	其中当 `(u,v)` 是曲面的正交参数 (即 `F -= 0`) 时, `overset ** =` 成立.
</p>

<p class="proof">
  先求矩阵
  <span class="formula">
    `(g^(alpha beta))`
    `= [E, F; F, G]^-1`
    `= 1/(E G-F^2) [G, -F; -F, E]`.
  </span>
  以 `Gamma_11^1` 为例:
  <span class="formula">
    `Gamma_11^1`
    `= 1/2 g^(xi 1) (2(del g_(1 xi))/(del u^1) - (del g_11)/(del u^xi))`
    `= 1/2 (2 {:g^11:}(del g_11)/(del u^1) - {:g^11:}(del g_11)/(del u^1)
    + 2 {:g^21:} (del g_12)/(del u^1) - {:g^21:} (del g_11)/(del u^2))`
    `= 1/(E G-F^2) (1/2 G E_u - F F_u + 1/2 F E_v)`.
  </span>
</p>

<p class="example">
	单位球面在球极投影参数下
	<span class="formula">
		` [E, F; F, G] = 4/(1+u^2+v^2) bm E`,<br/>
		  `[Gamma_11^1, Gamma_12^1; Gamma_21^1, Gamma_22^1]
		= 2/(1+u^2+v^2) [-u,-v; -v,u]`,<br/>
		  `[Gamma_11^2, Gamma_12^2; Gamma_21^2, Gamma_22^2]
	    = 2/(1+u^2+v^2) [v,-u; -u,-v]`.
	</span>
</p>

<h2>曲面的结构方程 (Gauss-Codazzi)</h2>

<h3>方程的推导</h3>

<p>	根据二阶连续可微函数的二阶偏导数可交换次序, 有
	<span class="formula">
		`bm r_(alpha beta) = bm r_(beta alpha)`,
		<span class="label" id="commu1"></span>
	</span>
	<span class="formula">
		`bm r_(alpha beta gamma) = bm r_(alpha gamma beta)`,
		<span class="label" id="commu2"></span>
	</span>
	<span class="formula">
		`bm n_(alpha beta) = bm n_(beta alpha)`.
		<span class="label" id="commu3"></span>
	</span>
</p>

<p>	先看 <a class="ref" href="#commu1"></a>.
  联系自然标架的运动方程, 它等价于 `Gamma_(alpha beta)`,
	`b_(alpha beta)` 关于 `alpha, beta` 对称.
</p>

<p>	再看 <a class="ref" href="#commu2"></a>. 因为
	<span class="formula">
		`bm r_(alpha beta gamma)`
		`= del/(del u^gamma)
		   (Gamma_(alpha beta)^xi bm r_xi + b_(alpha beta) bm n)`
		`= (del Gamma_(alpha beta)^xi)/(del u^gamma) bm r_xi
		   + Gamma_(alpha beta)^xi bm r_(xi gamma)
		   + (del b_(alpha beta))/(del u^gamma) bm n
       + b_(alpha beta) (del bm n)/(del u^gamma)`<br/>
		`= (del Gamma_(alpha beta)^xi)/(del u^gamma) bm r_xi
		   + Gamma_(alpha beta)^xi
		     (Gamma_(xi gamma)^eta bm r_eta + b_(xi gamma) bm n)`
		  `+ (del b_(alpha beta))/(del u^gamma) bm n
       + b_(alpha beta) (-b_gamma^xi bm r_xi)`<br/>
		`= ( ( del Gamma_(alpha beta)^xi)/(del u^gamma)
			 + Gamma_(alpha beta)^eta Gamma_(eta gamma)^xi
			 - b_(alpha beta) b_gamma^xi ) bm r_xi`
		  `+ ( Gamma_(alpha beta)^xi b_(xi gamma)
		   + (del b_(alpha beta))/(del u^gamma) ) bm n`,
	</span>
	而由 <a class="ref" href="#commu2"></a>
  知上式左边关于 `beta, gamma` 对称, 所以上式右边也是如此.
	利用 `bm r_1, bm r_2, bm n` 线性无关得到
  <b>Gauss 方程</b>:
	<span class="formula">
		`b_(alpha beta) b_gamma^xi
		- b_(alpha gamma) b_beta^xi`
		`= (del Gamma_(alpha beta)^xi)/(del u^gamma)
		- (del Gamma_(alpha gamma)^xi)/(del u^beta)`
		`+ Gamma_(alpha beta)^eta Gamma_(eta gamma)^xi
		- Gamma_(alpha gamma)^eta Gamma_(eta beta)^xi`,
	</span>
  和 <b>Codazzi 方程</b>:
	<span class="formula">
		` (del b_(alpha beta))/(del u^gamma)
		- (del b_(alpha gamma))/(del u^beta)`
    `= - Gamma_(alpha beta)^xi b_(xi gamma)
		+ Gamma_(alpha gamma)^xi b_(xi beta)`.
	</span>
</p>

<p>	最后看 <a class="ref" href="#commu3"></a>. 我们有
	<span class="formula">
		` bm n_(beta gamma)
		= del/(del u^gamma) (b_beta^xi bm r_xi)
		= (del b_beta^xi)/(del u^gamma) bm r_xi
		  + b_beta^xi(Gamma_(xi gamma)^eta bm r_eta + b_(xi gamma) bm n)`
		`= ( (del b_beta^xi)/(del u^gamma)
		  + b_beta^eta Gamma_(eta gamma)^xi ) bm r_xi
		  + b_beta^xi b_(xi gamma) bm n`.
	</span>
	同样由上式对 `beta, gamma` 对称有
	<span class="formula">
		` (del b_beta^xi)/(del u^gamma)
		- (del b_gamma^xi)/(del u^beta) =
		- b_beta^eta Gamma_(eta gamma)^xi
		+ b_gamma^eta Gamma_(eta beta)^xi`.
	</span>
  (由于 `(g^(eta xi))` 对称, `b_beta^xi b_(xi gamma) = b_(beta eta) g^(eta
  xi) b_(xi gamma)` 是对称矩阵的合同阵, 它关于 `beta, gamma`
  对称是显然的). 可以证明上式与 Codazzi 方程是等价的.
</p>

<p class="proof">
  计算
  <span class="formula">
    `g_(alpha xi) (
      (del b_beta^xi)/(del u^gamma)
    + b_beta^eta Gamma_(eta gamma)^xi
    )`
    `= (del b_(alpha beta))/(del u^gamma)
    - b_beta^xi (del g_(alpha xi))/(del u^gamma)
    + b_beta^eta Gamma_(alpha eta gamma)`
    `= (del b_(alpha beta))/(del u^gamma)
    + b_beta^xi (Gamma_(alpha xi gamma) - (del g_(alpha xi))/(del u^gamma)
    )`
    `= (del b_(alpha beta))/(del u^gamma)
    - b_beta^xi Gamma_(xi alpha gamma)`
    `= (del b_(alpha beta))/(del u^gamma)
    - Gamma_(alpha gamma)^xi b_(xi beta)`.
  </span>
  上式两边置换 `beta, gamma` 并相减, 就能联系起 Codazzi 方程的两个等价形式.
</p>

<h3>Gauss-Codazzi 方程中的独立方程</h3>

<p class="definition">
  由于 `alpha, beta, gamma, xi = 1, 2`, Gauss 方程共有 16 个, Codazzi
	方程共 8 个. 不过实质上 Gauss 方程只有 1 个独立方程, Codazzi 方程只有
	2 个独立方程. 为说明这一点, 用 `g_(delta xi)` 乘 Gauss 方程左边, 称为
  <b>Riemann 记号</b>:
	<span class="formula">
		`R_(delta alpha beta gamma)`
    `= g_(delta xi)(b_(alpha beta)b_gamma^xi - b_(alpha gamma)b_beta^xi)`
		`= b_(alpha beta) b_(gamma delta) - b_(alpha gamma) b_(beta delta)`
    `= |b_(alpha beta), b_(alpha gamma);
        b_(beta delta), b_(gamma delta)|`.
	</span>
</p>

<p class="corollary">
  <b>置换性质</b> Riemann 记号满足
  <span class="formula">
    ` R_(delta alpha beta gamma)`
    `= R_(beta gamma delta alpha)`
    `= -R_(alpha delta beta gamma)`
    `= -R_(delta alpha gamma beta)`.
  </span>
</p>

<p class="theorem">
  Gauss 方程的左边也满足同样的置换性质; 因此 Gauss 方程中只有一个独立方程
	<span class="formula">
		`R_1212 = -(b_11 b_22 - b_12^2)`.
	</span>
</p>

<ol class="proof">
  <li>
    利用 <a class="ref" href="#christoffel2"></a> 计算 Gauss
    方程左边一半的项:
    <span class="formula">
      `g_(delta xi) ((del Gamma_(alpha beta)^xi)/(del u^gamma)
      + Gamma_(alpha beta)^eta Gamma_(eta gamma)^xi)`
      `= (del Gamma_(delta alpha beta))/(del u^gamma)
      - (del g_(delta xi))/(del u^gamma) Gamma_(alpha beta)^xi
      + Gamma_(alpha beta)^eta Gamma_(delta eta gamma)`
      `= (del Gamma_(delta alpha beta))/(del u^gamma)
      + Gamma_(alpha beta)^eta (
        Gamma_(delta eta gamma)
        - (del g_(delta eta))/(del u^gamma)
      )`
      `= 1/2 (
        (del^2 g_(alpha delta))/(del u^gamma del u^beta)
      + (del^2 g_(beta delta))/(del u^gamma del u^alpha)
      - (del^2 g_(alpha beta))/(del u^gamma del u^delta)
      )`
      `- Gamma_(alpha beta)^eta Gamma_(eta delta gamma)`.
    </span>
  </li>
  <li>
    简记 `(del^2 g_(delta beta))/(del u^gamma del u^alpha)`
    `= (g_(delta beta))_(gamma alpha)`.
    置换 1 中的 `beta, gamma`, 令它们相减, 得
    <span class="formula">
      `1/2 (
        (g_(delta beta))_(gamma alpha)
      - (g_(alpha beta))_(gamma delta)
      - (g_(delta gamma))_(alpha beta)
      + (g_(alpha gamma))_(beta delta)
      )`
      `- |Gamma_(alpha beta)^eta, Gamma_(eta delta gamma);
          Gamma_(alpha gamma)^eta, Gamma_(eta delta beta)|`.
    </span>
    上式确实满足 Riemann 记号的置换性质.
  </li>
</ol>

<p class="theorem">
	Codazzi 方程的 `beta = gamma` 时, 为平凡等式. 于是 Codazzi
	方程只有两个独立方程
	<span class="formula">`{
		(del b_11)/(del u^2) - (del b_12)/(del u^1)
	  = b_(1 xi) Gamma_12^xi - b_(2 xi) Gamma_11^xi;
		(del b_21)/(del u^2) - (del b_22)/(del u^1)
	  = b_(1 xi) Gamma_22^xi - b_(2 xi) Gamma_21^xi;
	:}`</span>
</p>

<p class="corollary">
  当 `(u,v)` 为正交参数系, 即 `F -= 0` 时, Gauss 方程化简为
	<span class="formula">
		`-1/sqrt(EG) [(((sqrt E)_v)/sqrt G)_v + (((sqrt G)_u)/sqrt E)_u]
		= (LN-M^2)/(EG)`,
	</span>
	Codazzi 方程化简为
	<span class="formula">`{
		(L/sqrt E)_v - (M/sqrt E)_u = N(sqrt E)_v/G + M(sqrt G)_u/sqrt(E G);
    (N/sqrt G)_u - (M/sqrt G)_v = L(sqrt G)_u/E + M(sqrt E)_v/sqrt(E G);
	:}`</span>
  若 `(u,v)` 为单位正交标架, 两个方程可以继续简化, 见下文.
</p>

<h2>曲面的存在惟一性定理</h2>

<p class="theorem">
	设 `S: bm r(u^1,u^2)`, `bar S: bm r(u^1,u^2)` 是定义在同一个参数区域
	`D` 上的两个曲面, 如果 `S` 和 `bar S` 的第一, 第二基本形式在 `D`
	上处处相等, 则两个曲面相差一个 `E^3` 的刚体运动.
</p>

<p class="theorem">
	设平面区域 `D` 定义了两个二次微分式
	<span class="formula">
		`varphi = g_(alpha beta) "d"u^alpha"d"u^beta`,<br/>
		`psi = b_(alpha beta) "d"u^alpha"d"u^beta`,
	</span>
	且 `(b_(alpha beta))` 是对称阵, `(g_(alpha beta))` 是正定对称阵.
	如果 `Gamma_(alpha beta)^gamma`, `b_(alpha beta)`, `b_beta^alpha`
	满足 Gauss-Codazzi 方程, 则对 `D` 中任意的一点 `(u_0^1, u_0^2)`, 存在
	一个邻域 `U sube D` 及定义在 `U` 上的曲面 `bm r(u^1, u^2)`, 使得
	`varphi, psi` 分别为该曲面的第一, 第二基本形式.
</p>

<p class="remark">
	自然标架运动方程是一个一阶偏微分线性方程组, Gauss-Codazzi
	方程实质上是这个方程组的可积性条件.
</p>

<h2>正交活动标架</h2>

<h3>正交标架下的曲面运动方程</h3>

<p>	引入正交活动标架通常能简化计算.</p>

<p>	在曲面 `S` 各点的切平面上选取单位向量 `bm e_1, bm e_2`, 使
	`(:bm e_i, bm e_j:) = delta_(ij)`, `i, j = 1, 2`;
	再令 `bm e_3 = bm e_1 ^^ bm e_2`,
	则 `{bm r";" bm e_1, bm e_2, bm e_3}` 构成曲面的一个<b>正交标架</b>,
	或<b>规范标架</b>.
</p>

<p class="definition">
	定义参数区域 `D` 上的一阶微分形式 `omega_1, omega_2` 满足
	`"d"bm r = omega_1 bm e_1 + omega_2 bm e_2`, 则
	<span class="formula">
		`omega_1 = (:"d"bm r, bm e_1:)`,
		`omega_2 = (:"d"bm r, bm e_2:)`,<br/>
		`"I" = (:"d"bm r, "d"bm r:) = omega_1 omega_1 + omega_2 omega_2`.
	</span>
	又定义一阶微分形式 `omega_(ij)` 满足
  <span class="formula">
    `"d"bm e_i = sum_(j=1)^3 omega_(ij) bm e_j`,
  </span>
  换言之,
  <span class="formula">
    `omega_(i j) = (:"d"bm e_i, bm e_j:)`,
  </span>
  则由<a href="2.html#the-unit-vec">这个关于单位向量微分的定理</a>, 有
  `omega_(ij) + omega_(ji) = 0`, 故
	<span class="formula">
		`"II" = -(:"d"bm r, "d"bm e_3:)`
		`= -(:omega_1 bm e_1 + omega_2 bm e_2, omega_31 bm e_1 + omega_32
		bm e_2:)`
    `= omega_1 omega_13 + omega_2 omega_23`.
	</span>
	称
	<span class="formula">
		`"d"[bm r; bm e_1; bm e_2; bm e_3]
		= [omega_1, omega_2, 0;
		   0, omega_12, omega_13;
		   omega_21, 0, omega_23;
		   omega_31, omega_32, 0]
		  [bm e_1; bm e_2; bm e_3]`,
	</span>
  (其中 `omega_(ij) + omega_(ji) = 0`) 为<b>曲面正交标架的运动方程</b>.
</p>

<p class="corollary">
	由以上定义可以看出, 曲面的 `"I", "II"` 由正交标架运动方程的系数决定.
	且可以证明, `"I"` 与正交标架的选取无关; 法向量确定的情况下, `"II"`
	也与正交标架选取无关.
</p>

<h3>正交标架下的第一, 第二基本形式</h3>

<p>	由于 `bm r_u, bm r_v` 和 `bm e_1, bm e_2` 都是切平面的基, 可以设
	<span class="formula">
		`[bm r_u; bm r_v] = bm A [bm e_1; bm e_2]`, `bm A`
		为可逆矩阵.
	</span>
	故
	<span class="formula">
		`["d"u, "d"v] bm A = [omega_1, omega_2]`.
	</span>
	我们知道一阶微分形式 `omega_13, omega_23` 是 `"d"u, "d"v`
	的线性组合; 由上式, 它也可以表示为 `omega_1, omega_2` 的线性组合.
	故存在矩阵 `bm B` 使得
	<span class="formula">
		`[omega_13, omega_23] = [omega_1, omega_2] bm B`.
	</span>
</p>

<p> 现在来考虑曲面的第一, 第二基本形式. 我们有
	<span class="formula">
		` "I"
		= [omega_1, omega_2] [omega_1; omega_2]
		= ["d"u, "d"v] bm(A A)^T ["d"u; "d"v]`,</br>
		` "II"
		= [omega_13, omega_23][omega_1; omega_2]
		= [omega_1, omega_2] bm B [omega_1; omega_2]
		= ["d"u, "d"v] bm (A B A)^T ["d"u; "d"v]`.
	</span>
	联系 `"I", "II"` 的定义得
	<span class="formula">
		`[E,F; F,G] = bm(A A)^T`,
		`[L,M; M,N] = bm(A B A)^T`.
	</span>
</p>

<p class="corollary">
	上式重新得出了 `[E,F; F,G]` 正定的结论;
	由于对称矩阵在合同变换下保持对称, 所以 `bm B` 是对称矩阵.
</p>

<h3>正交标架下的 Weingarten 变换</h3>

<p> 由
	<span class="formula">
		`"d"bm r = omega_1 bm e_1 + omega_2 bm e_2`,<br/>
		`-"d"bm n = -"d"bm e_3 = omega_13 bm e_1 + omega_23 bm e_2`
		`= [omega_1, omega_2] bm B [bm e_1; bm e_2]`
	</span>
	以及 `cc W("d"bm r) = -"d"bm n` 知道, `bm B` 恰为 Weingarten
  变换在正交标架下的矩阵.
</p>

<p class="corollary">
	曲面的主曲率恰为 `bm B` 的两个特征值; Gauss 曲率等于 `|bm B|`;
  平均曲率等于 `1/2 "tr"bm B`.<br/>
  当曲面没有脐点时, 特别取 `bm e_1, bm e_2` 为曲面的主方向, 有
	<span class="formula">
		`(:cc W(bm e_1), bm e_1:) = k_1`,
		`(:cc W(bm e_2), bm e_2:) = k_2`,<br/>
		`(:cc W(bm e_1), bm e_2:) = (:cc W(bm e_2), bm e_1:) = 0`,
	</span>
	即 `bm B` 为对角矩阵. 这时 `"II" = k_1 omega_1 omega_1 + k_2 omega_2
	omega_2`.
</p>

<h2>正交标架下的曲面结构方程 (外微分法)</h2>

<h3>外微分形式</h3>

<p class="definition">
	称平面参数区域 `D = {(u, v)}` 上的函数为<b>零阶外微分形式</b>.
	零阶微分形式关于 `"d"u, "d"v` 的线性组合, 形如 `f"d"u + g"d"v`,
	称为<b>一阶外微分形式</b>. 如 `"d"u`, `"d"v` 都是一阶外微分形式.
</p>

<ol class="definition">
	类似向量外积, 定义两个一阶外微分形式间的<b>外积运算</b> `^^`, 满足
  <li>线性性: `(lambda varphi + mu psi) ^^ theta
    = lambda(varphi ^^ theta) + mu(psi ^^ theta)`;
  </li>
  <li>反交换律: `varphi ^^ psi =  - psi ^^ varphi`.</li>
  因此
	<span class="formula">
    `theta ^^ theta = 0`;<br>
		`(f_1"d"u + f_2"d"v) ^^ (g_1"d"u + g_2"d"v)`
    `= (f_1 g_2 - f_2 g_1) "d"u ^^ "d"v`.
	</span>
	可见任意两个一阶外微分形式的外积都形如 `f "d"u ^^ "d"v`,
  称为<b>二阶外微分形式</b>.
</ol>

<p class="definition">
	现在可以定义微分形式的<b>外微分运算</b> `"d"`.
	<span class="formula">
		` "d"f = (del f)/(del u) "d"u + (del f)/(del v) "d"v`,<br/>
		` "d"(f"d"u+g"d"v)`
    `= "d"f ^^ "d"u + "d"g ^^ "d"v`
		`= ((del g)/(del u) - (del f)/(del v)) "d"u ^^ "d"v`,<br/>
		` "d"(f "d"u ^^ "d"v) = "d"f ^^ "d"u ^^ "d"v = 0`.
	</span>
</p>

<p class="remark">
  记忆: 设 `varphi = f "d"u + g"d"v`, 则
	`"d"varphi = |del/(del u), del/(del v); f, g| "d"u ^^ "d"v`.
	曲面上区域 `D` 的 Green 公式可以写为
	<span class="formula">
		`oint_(del D) varphi = iint_D "d"varphi`.
	</span>
</p>

<ol class="property">
	设 `f`, `g` 是函数, `varphi` 是一阶微分形式, 则
  <li>`"d"(fg) = ("d"f)g + f("d"g)`;</li>
  <li>`"d"(f varphi) = "d"f ^^ varphi + f "d"varphi`;</li>
  <li>`"d"("d"f) = 0`.</li>
  其中最后一个公式只需把 `f_u, f_v` 代入 `"d"(f"d"u + g"d"v)` 中,
  利用混合偏导数相等即可.
</ol>

<h3>曲面的结构方程</h3>

<p>	对正交标架的曲面运动方程第一式
	`"d"bm r = omega_1 bm e_1 + omega_2 bm e_2` 两边求外微分,
	<span class="formula align">
    ` bb 0 = "d"(sum_(i=1)^2 omega_i bm e_i)`<br>
    `= sum_(i=1)^2 ("d"omega_i bm e_i - omega_i ^^ "d"bm e_i)`<br>
		`= sum_(i=1)^2 ("d"omega_i bm e_i - omega_i ^^ sum_(j=1)^3
      omega_(ij) bm e_j)`<br>
		`= sum_(i=1)^2 "d"omega_i bm e_i - sum_(j=1)^3 sum_(i=1)^2
			omega_i ^^ omega_(ij) bm e_j.`
	</span>
	由 `bm e_1, bm e_2, bm e_3` 线性无关知,
	<span class="formula">
		`"d"omega_j = sum_(i=1)^2 omega_i ^^ omega_(ij)`, `quad j = 1, 2`;
		<span class="label" id="for-omega-1-2"></span>
	</span>
	<span class="formula">
		`0 = sum_(i=1)^2 omega_i ^^ omega_(i3)`.
		<span class="label" id="for-omega-i3"></span>
	</span>
	将 `omega_(i3) = sum_(j=1)^2 h_(ij) omega_j` 代入 <a class="ref" href="#for-omega-i3"></a>, 得
	<span class="formula">
		` 0
		= sum_(i=1)^2 omega_i ^^ sum_(j=1)^2 h_(ij) omega_j
		= sum_(i=1)^2 sum_(j=1)^2 h_(ij) omega_i ^^ omega_j
		= (h_12 - h_21) omega_1 ^^ omega_2`.
	</span>
	所以 `h_12 = h_21`, <a class="ref" href="#for-omega-i3"></a>
  等价于矩阵 `bm B = (h_(ij))^T` 是对称的.
	另一方面, 因为 `omega_11 = omega_22 = 0`,
  <a class="ref" href="#for-omega-1-2"></a> 简化为
	<span class="formula">`{
		"d" omega_1 = omega_2 ^^ omega_21;
		"d" omega_2 = omega_1 ^^ omega_12;
	:}`.
	<span class="label" id="for-d-omega-1-2"></span>
	</span>
</p>

<p>	同样, 对运动方程的第二式
	`"d"bm e_i = sum_(j=1)^3 omega_(ij) bm e_j` (`i = 1, 2, 3`) 求外微分,
	<span class="formula align">
		` bb 0 = "d"(sum_(j=1)^3 omega_(ij) bm e_j)`<br>
    `= sum_(j=1)^3 ("d"omega_(ij) bm e_j - omega_(ij) ^^ "d"bm e_j)`<br>
    `= sum_(j=1)^3 ("d"omega_(ij) bm e_j
    - omega_(ij) ^^ sum_(k=1)^3 omega_(jk) bm e_k)`<br>
		`= sum_(j=1)^3 "d"omega_(ij) bm e_j
    - sum_(k=1)^3 sum_(j=1)^3 omega_(ij) ^^ omega_(jk) bm e_k`<br>
		`= sum_(k=1)^3 ("d"omega_(ik)
		  - sum_(j=1)^3 omega_(ij) ^^ omega_(jk)) bm e_k`.
	</span>
	因此
	<span class="formula">
		`"d"omega_(ik) = sum_(j=1)^3 omega_(ij) ^^ omega_(jk)`,
		`quad i, k = 1, 2, 3`.
	</span>
	由于 `omega_(ik)` 的反对称性, `i = k` 时, 上面的等式是平凡的;
	将 `i != k` 的情形写出, 即
	<span class="formula">
		(Gauss) `"d"omega_12 = omega_13 ^^ omega_32`;
		<span class="label" id="for-otho-gauss"></span>
	</span>
	<span class="formula">
	(Codazzi) `{
		"d"omega_13 = omega_12 ^^ omega_23;
		"d"omega_23 = omega_21 ^^ omega_13;
	:}`.
		<span class="label" id="for-otho-codazzi"></span>
	</span>
	<a class="ref" href="#for-d-omega-1-2"></a>,
  <a class="ref" href="#for-otho-gauss"></a>,
  <a class="ref" href="#for-otho-codazzi"></a>
  合称为曲面<b>正交标架的结构方程式</b>,
	它们是正交标架运动方程的可积性条件.
</p>

<p>
  下面说明 <a class="ref" href="#for-otho-gauss"></a> 就是 Gauss 方程,
  <a class="ref" href="#for-otho-codazzi"></a> 就是 Codazzi 方程.
</p>

<p class="proof">
	设 `(u,v)` 是曲面的正交参数, 取 `bm e_1 = bm r_u/sqrt E`, `bm e_2 = bm
	r_v/sqrt G`, 容易验证 `bm e_1, bm e_2` 是切平面的单位正交基.
	此时
	<span class="formula">
		`bm A = [sqrt E, 0; 0, sqrt G]`,<br/>
		`bm B = bm A^-1 [L,M; M,N] (bm A^T)^-1
		      = [L//E, M//sqrt(EG); M//sqrt(EG),N//G]`.
	</span>
	于是 `omega_1 = sqrt E "d"u`, `omega_2 = sqrt G"d"v`.
	利用 <a class="ref" href="#for-d-omega-1-2"></a> 有
	<span class="formula">
		` omega_12 ^^ omega_2 = "d"omega_1 = -(sqrt E)_v "d"u ^^ "d"v
		= -(sqrt E)_v/sqrt G "d"u ^^ omega_2`,<br/>
		` omega_21 ^^ omega_1 = "d"omega_2 = (sqrt G)_u "d"u ^^ "d"v
		= -(sqrt G)_u/sqrt E "d"v ^^ omega_1`.
	</span>
	记 `beta = -(sqrt E)_v/sqrt G "d"u`, `gamma = (sqrt G)_u/sqrt E "d"v`,
	由上式,
	<span class="formula">
		`(omega_12 - beta - gamma) ^^ omega_1 = (omega_12 - gamma) ^^
		omega_1 = 0`,<br/>
		`(omega_12 - beta - gamma) ^^ omega_2 = (omega_12 - beta) ^^
		omega_2 = 0`.
	</span>
	但 `omega_1 ^^ omega_2 != 0`, 所以 `omega_12 - beta - gamma = 0`, 即
	<span class="formula">
		`omega_12 = beta + gamma = -(sqrt E)_v/sqrt G "d"u + (sqrt
		G)_u/sqrt E "d"v`.
    <span class="label" id="for-omega-12"></span>
	</span>
	再利用 `[omega_13, omega_23] = [omega_1, omega_2] bm B` 得
	<span class="formula">
		`omega_13 = L/sqrt E "d"u + M/sqrt E "d"v`,<br/>
		`omega_23 = M/sqrt G "d"u + N/sqrt G "d"v`.
	</span>
	将上述结果代入 <a class="ref" href="#for-otho-gauss"></a> 计算,
  就得到正交参数下的 Gauss 方程;
	代入 <a class="ref" href="#for-otho-codazzi"></a>,
  就得到正交参数下的 Codazzi 方程.
</p>

<p class="remark">
	Gauss 方程 <a class="ref" href="#for-otho-gauss"></a> 也可以写为
	<span class="formula">
		` "d"omega_12
		= -sum_(i=1)^2 h_(1i) omega_i ^^ sum_(j=1)^2 h_(2j) omega_j
		= -sum_(i=1)^2 sum_(j=1)^2 h_(1i) h_(2j) omega_i ^^ omega_j`
		`=-(h_11 h_22 - h_12 h_21) omega_1 ^^ omega_2
		= -K omega_1 ^^ omega_2`.
	</span>
</p>

<h3>曲面的几何量</h3>

<p>	设 `{bm r";" bm e_1, bm e_2, bm e_3}` 和 `{bm bar r";" bm bar e_1, bm
  bar e_2, bm bar e_3}` 是曲面的两组正交标架 (都是右手系),
	`{omega_1, omega_2, omega_12, omega_13, omega_23}` 和
	`{bar omega_1, bar omega_2, bar omega_12, bar omega_13, bar omega_23}`
	是相应的诸微分形式. 又设这两个标架间相差一个 `theta` 角的旋转:
	<span class="formula">
    `[bm bar e_1; bar bm e_2] = bm T [bm e_1; bm e_2]`,
    `quad bm T = [cos theta, -sin theta; sin theta, cos theta]`,
	</span>
	计算知 `{bar omega_1, bar omega_2}` 与 `{omega_1, omega_2}` 之间,
	`{bar omega_13, bar omega_23}` 与 `{omega_13, omega_23}`
	之间均相差一个相同的旋转:
	<span class="formula">
    `[bar omega_1; bar omega_2] = bm T [omega_1; omega_2]`,
    `quad [bar omega_13; bar omega_23] = bm T [omega_13; omega_23]`.
	</span>
  由此可以验证以下的量与正交标架选取无关. 这些不依赖于 (同向)
  正交标架选取的量称为<b>曲面的几何量</b>:
	<ol>
		<li>第一基本形式 `"I" = omega_1 omega_1 + omega_2 omega_2`;</li>
		<li>第二基本形式 `"II" = omega_1 omega_13 + omega_2 omega_23`;</li>
		<li>第三基本形式 `"III" = omega_13 omega_13 + omega_23 omega_23`;
		</li>
		<li>面积元 `"d"A = omega_1 ^^ omega_2`;</li>
		<li>Gauss 映射的面积元 `"d"sigma = omega_13 ^^ omega_23 =
			K"d"u^^"d"v`;
		</li>
		<li>Hopf 不变式 `psi = omega_1 omega_23 - omega_2 omega_13`.</li>
	</ol>
</p>

<p class="remark">
  `omega_12` 称为<b>联络形式</b>,
  我们将在下一章研究曲面的内蕴几何学时进一步讨论.
	由 `bar omega_12 = (:"d"bm bar e_1, bm bar e_2:)` 计算知
	<span class="formula">
		`bar omega_12 = omega_12 + "d"theta`.
	</span>
	故 `omega_12` 依赖于正交标架的选取, 不是几何量.
</p>

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